through Q on the ellipsoidal surface. It must cut the arc A'C'A at some point between U3 and U4, arid, if continued on and on, it must cut the ellipse ACA'O'A successively between Ui and U3, or between U3 and U4 ; never between U2 and U3, or U4 and Uj. This, for the extreme case of the smallest axis zero, is illustrated by the path IQQ'Q^Q"' Q,VQV in fig. 2.
§ 32. If now, on the other hand, we commence a geodetic through any point J betw een Ui and U4, or between U2 and 1J3, it will never cut the principal section containing the umbilicus, either between U, and U2 or between U3 and U4. This, for the extreme case of CC^O, is illustrated in fig. 3.
Fig. 2.
§ 33. Tt seems not improbable that if the figure deviates by ever so little from being exactly ellipsoidal, Maxwell’s condition might be fulfilled. It seems indeed quite probable that Maxwell's condition (see §§ 13, 29, above) is fulfilled by a geodetic on a closed surface of any shape in general, and that exceptional cases, in which the question of §29 is to be answered in the negative, are merely particular surfaces of definite shapes, infinitesimal deviations from which will allow the question to be answered in the affirmative.
§ 34. Now with an affirmative answer to the question—is Maxw'ell's condition fulfilled ?—what does the Boltzmann-Maxwell doctrine assert in respect to a geodetic on a closed surface ? The mere wording of Maxwell's statement, quoted in §13 above, is not applicable to this case, but the meaning
of the doctrine as interpreted from previous writings both of Boltzmann and Maxwell, and subsequent writings of Boltzmann, and of Rayleigh *, the most recent supporter of the doctrine, is that a single geodetic drawn long enough will not only fulfil Maxwell's condition of passing infinitely near to every point of the surface in all directions, but will pass with equal frequencies in all directions ; and as many times within a certain infinitesimal distance +8 of any one point P as of any other point P' anywhere over the whole surface. This, if true, would be an exceedingly interesting theorem*
§ 35. I have made many efforts to test it for the case in which the closed surface is reduced to a plane with other boundaries than an exact ellipse (for which, as we have seen
Fig, 3.
enclosed area turned on meeting the boundary, according to the law of equal angles of incidence and reflection, which corresponds also to the case of an ideal perfectly smooth; non-rotating billiard-ball moving in straight lines except when it strikes the boundary of the table ; the boundary being of any shape whatever, instead of the ordinary rectangular boundary of an ordinary billiard-table, and bein# perfecti}' elastic. An interesting illustration, easily seen
* Phil. Mag., January 1C00.
